recognition_deficit
The recognition deficit theorem establishes that J-cost is strictly positive for every positive real number different from one. Cross-domain researchers in Recognition Science cite it as the shared foundation for non-equilibrium costs in turbulence, disease, game theory, and neurodevelopment. The proof reduces directly to the core positivity lemma for J-cost via a single function application.
claim$forall r : mathbb{R}, 0 < r to r neq 1 to 0 < J(r)$, where $J$ is the J-cost function.
background
The module C16 establishes J-cost positivity universality as the off-equilibrium analogue of C7. RecognitionDeficit is the proposition forall r in R, 0 < r implies r neq 1 implies 0 < Jcost r. This single statement is specialized across seven domains: turbulent flow, disease deviation from homeostasis, off-target CRISPR effects, off-equilibrium games, market arbitrage, biased reasoning, and recognition deficit in neurodevelopment.
proof idea
The proof is a term-mode one-liner that applies the upstream lemma Jcost_pos_of_ne_one to the supplied r, positivity hypothesis, and inequality hypothesis, directly discharging the universal quantifier.
why it matters in Recognition Science
This theorem supplies the recognition deficit case inside jPositivityUniversalityCert, which aggregates the seven domain instances into one certificate. It completes the cross-domain extension of C7, confirming that the same J-cost positivity source (proved by rewriting as a square and using positivity of squares) governs all listed specializations. The result sits in the Recognition Science forcing chain after J-uniqueness and before domain-specific applications.
scope and limits
- Does not compute explicit numerical values of J-cost for specific r.
- Does not apply when r is less than or equal to zero.
- Does not cover the equilibrium case where J-cost equals zero at r equals one.
- Does not derive the closed-form expression of the J-cost function.
formal statement (Lean)
51theorem recognition_deficit : RecognitionDeficit :=
proof body
Term-mode proof.
52 fun r hr hne => Jcost_pos_of_ne_one r hr hne
53
54/-! ## Universality: all seven are definitionally the same proposition. -/
55